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How many geometric progressions is/are possible containing 27,8 and 12 as three of its/their terms?

Options

  1. AOne
  2. BTwo
  3. CFour
  4. DInfinitely many

Correct answer

D. Infinitely many

Step-by-step solution

Let ' a ' be the first term & ' x ' be the common ratio. Also, suppose 27,8 & 12 be the p ^ th , q ^ th & r ^ th term of the G.P. aligned ax ^ p -1 & =27 ax ^ q -1 & =8 & ax ^ r -1 & =12 aligned Now, 27 8^2=12^3 aligned & ax ^ p -1 x ( ax ^ q -1 )^2= ( ax ^ r -1 )^3 & x ^ p -1 x ^ 2 q -2 = x ^ 2 r -3 & p -1+2 q -2=3 r -3 aligned p+2 q-3 r=0 ...(1) There are infinitely many solutions for the eq. (1).

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